Problem: Nuan is using a faulty barcode scanner that misreads barcodes $5\%$ of the time. Let $B$ = the number of barcodes Nuan scans until the scanner misreads one. Assume the success of each scan is independent. Find the probability that Nuan scans greater than $10$ barcodes before the scanner misreads one. You may round your answer to the nearest hundredth. $P(B>10)=$
Answer: Without a fancy calculator For each barcode scanned: $P({\text{misread}})=0.05$ $P(\text{read correctly}})=0.95$ If Nuan scans greater than $10$ barcodes before the scanner misreads one, then the scanner must read the first $10$ barcodes correctly. $\begin{aligned} P(B>10)&=P(10\text{ read correctly}) \\\\ &=(0.95})^{10} \\\\ &\approx 0.5987 \end{aligned}$ $P(B>10)\approx 0.5987 \approx 0.6$